Nucl.Phys.A 1037(2023)122687 Accounting for the LHCb observation that the reported hidden-charm strange pentaquark $P_{cs}(4459)^0$ can split into two substructures, $P_{cs}(4455)^0$ and $P_{cs}(4468)^0$, with a mass difference of 13 MeV as well as the newly observed hidden-charm pentaquark resonance $P_{cs}(4338)^0$ with strangeness, we study within the double-peak scenario for the $P_{cs}(4459)^0$ state the near-threshold $J/\psi$ meson production from protons and nuclei by considering incoherent direct non-resonant (${K^-}p \to {J/\psi}\Lambda$) and two-step resonant (${K^-}p \to P_{csi}^0 \to {J/\psi}\Lambda$, $i=1$, 2, 3; $P_{cs1}^{0}=P_{cs}(4338)^0$, $P_{cs2}^{0}=P_{cs}(4455)^0$, $P_{cs3}^{0}=P_{cs}(4468)^0$) charmonium production processes with the main goal of clarifying the possibility to observe within this scenario both above two substructures contributing to the $P_{cs}(4459)^0$ state and the $P_{cs}(4338)^0$ resonance in this production. We calculate the absolute excitation functions, energy and momentum distributions for the non-resonant, resonant and for the combined (non-resonant plus resonant) production of $J/\psi$ mesons on protons as well as on carbon and tungsten target nuclei at near-threshold incident antikaon beam momenta by assuming the spin-parity assignments of the hidden-charm resonances $P_{cs}(4338)^0$, $P_{cs}(4455)^0$ and $P_{cs}(4468)^0$ as $J^P=(1/2)^-$, $J^P=(1/2)^-$ and $J^P=(3/2)^-$ within four different realistic choices for the branching ratios of their decays to the ${J/\psi}\Lambda$ mode (0.125, 0.25, 0.5 and 1\%) as well as for two options for the branching fraction of their decays to the $K^-p$ channel (0.01 and 0.001\%). We show that these combined observables reveal clear sensitivity to these scenarios. Hence, they may be an important tool to provide further evidence for the existence of the above strange hidden-charm pentaquark resonances.